If `f:R in R` be such that
f(x)`=sqrt(sin(cosx))+"In"(-2cos^(2) x+3 cos x-1)`, then `int_(x_(1))^(x_(2)) [cos x-(1)/(2)]dx` is equal to, where `x1.x2 in D` and [.] denotes the greatest integer function,

Options

A. 0
B. `(1)/(2)(x_(2)-x_(1))`
C. `x_(1)-x_(2)`
D. `(1)/(2)(x_(1)-x_(2))`

Answer

A

Explanation

Clearly, `sqrt(sin(cos x))` is defined for all `x in R`.
In `(-2 cos^(2)x+3 cos x-1)` is defined, if
`-2cos^(2)x+3 cos x-1 gt 0`
`rArr 2 cos^(2)x-3 cos x+1 lt 0`
`rArr (2 cos x-1)(cos x-1 lt 0`
`rArr (1)/(2) lt cos x lt 1`
`rArr 0 lt cos x-(1)/(2) lt (1)/(2) rArr [cos x-(1)/(2)]=0`
`:. overset(x2)underset(x1)int [cos x-(1)/(2)]dx=overset(x2)underset(x1)int 0 dx=0`